Chapter 3Parameter Selection Methods in InverseProblem Formulation

H.T. Banks, Ariel Cintron-Arias, and Franz Kappel

Abstract We discuss methods for a priori selection of parameters to be estimatedin inverse problem formulations (such as Maximum Likelihood, Ordinary andGeneralized Least Squares) for dynamical systems with numerous state variablesand an even larger number of parameters. We illustrate the ideas with an in-hostmodel for HIV dynamics which has been successfully validated with clinical dataand used for prediction and a model for the reaction of the cardiovascular system toan ergometric workload.

H.T. Banks (�)Center for Research in Scientific Computation and Center for Quantitative Sciencesin Biomedicine, Department of Mathematics, North Carolina State University, Raleigh,NC 27695-8212e-mail: htbanks@ncsu.edu

A. Cintron-AriasCenter for Research in Scientific Computation and Center for Quantitative Sciencesin Biomedicine, Department of Mathematics, North Carolina State University, Raleigh,NC 27695-8212

Department of Mathematics and Statistics, East Tennessee State University, Johnson City,TN 37614-0663e-mail: cintronarias@mail.etsu.edu

F. KappelCenter for Research in Scientific Computation and Center for Quantitative Sciencesin Biomedicine, Department of Mathematics, North Carolina State University, Raleigh,NC 27695-8212

Institute for Mathematics and Scientific Computation, University of Graz, A8010 Graz, Austriae-mail: franz.kappel@uni-graz.at

J.J. Batzel et al. (eds.), Mathematical Modeling and Validation in Physiology,Lecture Notes in Mathematics 2064, DOI 10.1007/978-3-642-32882-4 3,© Springer-Verlag Berlin Heidelberg 2013

43

44 H.T. Banks et al.

3.1 Introduction

There are many topics of great importance and interest in the areas of modeling andinverse problems which are properly viewed as essential in the use of mathematicsand statistics in scientific inquiries. A brief, noninclusive list of topics include theuse of traditional sensitivity functions (TSF) and generalized sensitivity functions(GSF) in experimental design (what type and how much data is needed, where/whento take observations) [9–11, 16, 56], choice of mathematical models and theirparameterizations (verification, validation, model selection and model comparisontechniques) [8, 12, 13, 17, 21–24, 41], choice of statistical models (observationprocess and sampling errors, residual plots for statistical model verification, use ofasymptotic theory and bootstrapping for computation of standard errors, confidenceintervals) [8, 14, 30, 31, 54, 55], choice of cost functionals (maximum likelihoodestimation, ordinary least squares, weighted least squares, generalized least squares,etc.) [8, 30], as well as parameter identifiability and selectivity. There is extensiveliterature on each of these topics and many have been treated in surveys in one formor another ([30] is an excellent monograph with many references on the statisticallyrelated topics) or in earlier lecture notes [8].

We discuss here an enduring major problem: selection of those model parameterswhich can be readily and reliably (with quantifiable uncertainty bounds) estimatedin an inverse problem formulation. This is especially important in many areas ofbiological modeling where often one has large dynamical systems (many statevariables), an even larger number of unknown parameters to be estimated anda paucity of longitudinal time observations or data points. As biological andphysiological models (at the cellular, biochemical pathway or whole organism level)become more sophisticated (motivated by increasingly detailed understanding—orlack thereof—of mechanisms), it is becoming quite common to have large systems(10–20 or more differential equations), with a plethora of parameters (25–100) butonly a limited number (50–100 or fewer) of data points per individual organism.For example, we find models for the cardiovascular system [16, Chap. 1] (wherethe model has 10 state variables and 22 parameters) and [50, Chap. 6] (where themodel has 22 states and 55 parameters), immunology [48] (8 states, 24 parameters),metabolic pathways [32] (8 states, 35 parameters) and HIV progression [7, 42].Fortunately, there is a growing recent effort among scientists to develop quantitativemethods based on sensitivity, information matrices and other statistical constructs(see, for example, [9–11, 25, 27, 37, 38, 59]) to aid in identification or parameterestimation formulations. We discuss here one approach using sensitivity matricesand asymptotic standard errors as a basis for our developments. To illustrate ourdiscussions, we will use two models from the biological sciences: (a) a recentlydeveloped in-host model for HIV dynamics which has been successfully validatedwith clinical data and used for prediction [3, 7]; (b) a global non-pulsatile modelfor the cardiovascular system which has been validated with data from bicycleergometer tests [16, 44].

3 Parameter Selection Methods in Inverse Problem Formulation 45

The topic of system and parameter identifiability is actually an old one. In thecontext of parameter determination from system observations or output it is atleast 40 years old and has received much attention in the peak years of linearsystem and control theory in the investigation of observability, controllability anddetectability [6, 18, 19, 35, 39, 43, 46, 52, 53]. These early investigations and resultswere focused primarily on engineering applications, although much interest in otherareas (e.g., oceanography, biology) has prompted more recent inquiries for bothlinear and nonlinear dynamical systems [5, 15, 29, 34, 40, 47, 57, 59–61].

3.2 Statistical Models for the Observation Process

One has errors in any data collection process and the presence of these errorsis reflected in any parameter estimation result one might obtain. To understandand treat this, one usually specifies a statistical model for the observation processin addition to the mathematical model representing the dynamics. To illustrateideas here we use ordinary least squares (OLS) consistent with an error model forabsolute error in the observations. For a discussion of other frameworks (maximumlikelihood in the case of known error distributions, generalized least squaresappropriate for relative error models) see [8].

In order to be more specific we assume that the dynamics of the system ismodeled by a system of ordinary differential equations:

Px.t/ D g.t; x.t/; �/; t � t0; x.t0/ D x0.�/; (3.1)

z.t/ D h.t; x.t/; �/; t � t0; � 2 A ; (3.2)

where G � Rn and A � R

p are open sets and g W Œt0; 1/ � G � A ! Rn, x0 W

A ! Rn and h W Œt0; 1/ � G � A ! R are sufficiently smooth functions. The set

A is called the set of admissible parameters and z.�/ is the measurable output ofthe system, which for simplicity we assume to be scalar. Let x.t/ D x.t I �/ denotethe solution of (3.1) for given � 2 A and set

f .t; �/ D h.t; x.t I �/; �/; t � t0; � 2 A :

Thenz.t/ D f .t I �/; t � t0; � 2 A ; (3.3)

is the output model corresponding to the model (3.1), (3.2). It is clear that an outputmodel of the form (3.3) can also originate from dynamical models, where insteadof the ODE-system (3.1) we may have a system of delay equations or some partialdifferential equation. In order to describe the observation process we assume thereexists a vector �0 2 A , referred to as the true or nominal parameter vector, forwhich the output z.t/ D f .t; �0/ describes the output of the real system exactly. Atgiven sampling times

t0 � t1 < � � � < tN ;

46 H.T. Banks et al.

we have measurements yj for the output of the real system, j D 1; : : : ; N . It is alsoreasonably assumed that each of the N longitudinal measurements yj is affected byrandom deviations �j from the true underlying output. That is, we assume that themeasurements are given by

yj D f .tj I �0/ C �j ; j D 1; : : : ; N: (3.4)

The measurement errors �j are assumed to be realizations of random variablesEj satisfying the following assumptions:

(i) The errors Ej have mean zero, E�Ej

� D 0.(ii) The errors Ej have finite common variance, var.Ej / D �2

0 < 1.(iii) the errors Ej are independent (i.e., cov.Ej ; Ek/ D 0 whenever j ¤ k) and

identically distributed.

According to (3.4) the measurements yj are realizations of random variables Yj ,the observations at the sampling times tj . Then the statistical model for the scalarobservation process is

Yj D f .tj I �0/ C Ej ; j D 1; : : : ; N: (3.5)

Assumptions (i)–(iii) imply that the mean of the observations is equal to the modeloutput for the nominal parameter vector, E

�Yj

� D f .tj I �0/, and the variance in theobservations is constant in time, var.Yj / D �2

0 , j D 1; : : : ; N .

For given measurements y D .y1; : : : ; yN /T the estimate O�OLS for �0 is obtainedby minimizing

J.y; �/ DNX

j D1

�yj � f .tj I �/

�2 D jy � F.�/j2 D jF.�/ � F.�0/ � �j2; (3.6)

where we have set

F.�/ D �f .t1I �/; : : : ; f .tN I �/

�T; � D �

�1; : : : ; �N

�T;

and j � j is the Euclidean norm in RN . The estimate O�OLS is a realization of a random

variable, the least squares estimator �OLS. In order to indicate the dependenceon N we shall write O�N

OLS and �NOLS when needed. From [54] we find that under a

number of regularity and sampling conditions, as N ! 1, �NOLS is approximately

distributed according to a multivariate normal distribution, i.e.,

�NOLS � Np

��0; ˙N

0

�; (3.7)

where ˙N0 D �2

0

�N˝0

��1 2 Rp�p and

˝0 D limN !1

1

N�N .�0/

T�N .�0/:

3 Parameter Selection Methods in Inverse Problem Formulation 47

The N � p matrix �N .�/ is known as the sensitivity matrix of the system, and isdefined as

�N .�0/ D�@f .ti I �0/

@�j

�

1�i�N; 1�j �pD @F

@�.�0/ D r� F.�0/: (3.8)

Asymptotic theory [8, 30, 54] requires existence and non-singularity of ˝0. Thep � p matrix ˙N

0 is the covariance matrix of the estimator �N .If the output model (3.3) corresponds to the model (3.1), (3.2) then the derivatives

of f with respect to the parameters are given by

@f

@�j

.t; �/ D @h

@x.t; x.t I �/; �/

@x

@�j

.t I �/ C @h

@�j

.t; x.t I �/; �/; j D 1; : : : ; p;

(3.9)where w.t I �/ D �

.@x=@�1/.t I �/; : : : ; .@x=@�p/.t I �/� 2 R

N �p is obtained bysolving

Px.t I �/ D g.t; x.t I �/; �/; x.t0I �/ D x0.�/;

Pw.t; �/ D @g

@x

�t; x.t I �/; �

�w.t I �/ C @g

@�.t; x.t I �/; �/; w.t0I �/ D @x0

@�.�/;

(3.10)

from t D t0 to t D tN . One could alternatively obtain the sensitivity matrix usingdifference quotients (usually less accurately) or by using automated differentiationsoftware (for additional details on sensitivity matrix calculations, see [8, 9, 27, 28,33, 36]).

The estimate O�OLS D O�NOLS is a realization of the estimator �OLS, and is calculated

using a realization fyigNiD1 of the observation process fYigN

iD1, while minimiz-ing (3.6) over � . Moreover, the estimate O�OLS is used in the calculation of thesampling distribution for the parameters. The generally unknown error variance �2

0

is approximated by

O�2OLS D 1

N � p

NX

j D1

�yj � f .tj I O�N

OLS/�2

; (3.11)

while the covariance matrix ˙N0 is approximated by

O NOLS D O�2

OLS

��N . O�OLS/T�N . O�OLS/

��1

: (3.12)

As discussed in [8, 30, 54] an approximate for the sampling distribution of theestimator is given by

�OLS D �NOLS � Np.�0; ˙N

0 / Np. O�NOLS; O N

OLS/: (3.13)

48 H.T. Banks et al.

Asymptotic standard errors can be used to quantify uncertainty in the estimation,and they are calculated by taking the square roots of the diagonal elements of thecovariance matrix O N

OLS, i.e.,

SEk. O�NOLS/ D

q. O N

OLS/k;k; k D 1; : : : ; p: (3.14)

Before describing the algorithm in detail and illustrating its use, we provide somemotivation underlying the use of the sensitivity matrix �.�0/ of (3.8) and the Fisherinformation matrix F .�0/ D �

1=�20

��T.�0/�.�0/. Both of these matrices play a

fundamental role in the development of the approximate asymptotic distributionaltheory resulting in (3.13) and (3.14). Since a prominent measure of the ability toestimate a parameter is related to its associated standard errors in estimation, it isworthwhile to briefly outline the underlying approximation ideas in the asymptoticdistributional theory.

Ordinary least squares problems involve choosing � D �OLS to minimizethe difference between observations Y and model output F.�/, i.e., minimizejY � F.�/j. However the approximate asymptotic distributional theory (e.g., see[54, Chap. 12]) which is exact for model outputs linear in the parameters, employs afundamental linearization in the parameters in a neighborhood of the hypothesized“true” parameters �0. Replacing the model output with a first order linearizationabout �0, we then may seek to minimize for � in the approximate functional

ˇY � F.�0/ � r� F.�0/.� � �0/

ˇ:

If we use the statistical model Y D F.�0/ C E and let ı� D � � �0, we thus wishto minimize

ˇˇE � �.�0/ı�

ˇˇ;

where �.�0/ D r� F.�0/ is the N � p sensitivity matrix defined in (3.8). Thisis a standard optimization problem [45, Sect. 6.11] whose solution can be givenusing the pseudo inverse �.�0/

� defined in terms of minimal norm solutionsof the optimization problem and satisfying �.�0/� D .�.�0/

T�.�0//��.�0/T D

�20 F .�0/

��.�0/T. The solution is

ı� D �.�0/�E

or

�LIN D �0 C �.�0/�E D �0 C �2

0 F .�0/��.�0/TE :

If F .�0/ is invertible, then the solution (to first order) of the OLS problem is

�OLS �LIN D �0 C �20 F .�0/

�1�.�0/TE : (3.15)

3 Parameter Selection Methods in Inverse Problem Formulation 49

This approximation, for which the asymptotic distributional theory is exact, can bea reasonable one for use in developing an approximate nonlinear asymptotic theoryif ı� is small, i.e., if the OLS estimated parameter is close to �0.

From these calculations, we see that the rank of �.�0/ and the conditioning (orill-conditioning) of F .�0/ play a significant role in solving OLS inverse problemsas well as in any asymptotic standard error formulations based on this linearization.Observe that the error (or noise) E in the data will in general be amplified as theill-conditioning of F increases. We further note that the N � p sensitivity matrix�.�0/ is of full rank p if and only if the p � p Fisher matrix F .�0/ has rank p,or equivalently, is nonsingular. These underlying considerations have motivated anumber of efforts (e.g., see [9–11]) on understanding the conditioning of the Fishermatrix as a function of the number N and longitudinal locations ftj gN

j D1 of datapoints as a key indicator for well-formulated inverse problems and as a tool inoptimal design, especially with respect to computation of uncertainty (standarderrors, confidence intervals) for parameter estimates.

In view of the considerations above (which are very local in nature—boththe sensitivity matrix and the Fisher information matrix are taken at the nominalvector �0), one should be pessimistic about using these quantities to obtain anynonlocal selection methods or criteria for estimation. Indeed, for nonlinear complexsystems, it is easy to argue that questions related to some type of global parameteridentifiability are not fruitful questions to pursue.

3.3 Subset Selection Algorithm

The focus of our presentation here is how one chooses a priori (i.e., before anyinverse problem calculations are carried out) which parameters can be readilyestimated with a typical longitudinal data set. We illustrate an algorithm, developedrecently in [27], to select parameter vectors that can be estimated from a givendata set using an ordinary least squares inverse problem formulation (similar ideasapply if one is using a relative error statistical model and generalized least squaresformulations). Let q 2 R

p0 be the parameter vectors being at our disposal forparameter estimation and denote by q0 2 R

p0 the vector of the correspondingnominal values. Given a number p < p0 of parameters we wish to identify,the algorithm searches all possible choices of p different parameters among thep0 parameters and selects the one which is identifiable (i.e., the correspondingsensitivity matrix has full rank p) and minimizes a given uncertainty quantification(e.g. by means of asymptotic standard errors). Prior knowledge of a nominal setof values for all parameters along with the observation times for data (but not thevalues of the observations) will be required for our algorithm. For p < p0 we set

Sp D ˚� 2 R

pj � is a sub-vector of q 2 Rp0

�;

50 H.T. Banks et al.

i.e., � 2 Sp is given as � D �qi1 ; : : : ; qip

�Tfor some 1 � i1 < � � � < ip � p0.

The corresponding nominal vector is �0 D �.q0/i1 ; : : : ; .q0/ip

�T.

As we have stated above, to apply the parameter subset selection algorithmwe require prior knowledge of nominal variance and nominal parameter values.These nominal values of �0 and �0 are needed to calculate the sensitivity matrix,the Fisher matrix and the corresponding covariance matrix defined in (3.12). Forour illustration presented in Sect. 3.5, we use the variance and parameter estimatesobtained in previous investigations of the models as nominal values. In problemsfor which no prior estimation has been carried out, one must use knowledge of theobservation process error and some knowledge of viable parameter values that mightbe reasonable with the model under investigation.

The uncertainty quantification we shall use is based on the considerations givenin the previous section. Let � 2 R

p be given. As an approximation to the covariancematrix of the estimator for � we take

˙.�0/ D �20

��.�0/T�.�0/

��1 D F .�0/�1:

We introduce the coefficients of variation for �

�i .�0/ D�˙.�0/i;i

�1=2

.�0/i

; i D 1; : : : ; p; (3.16)

and take as a uncertainty quantification for the estimates of � the selection scoregiven by the Euclidean norm in � 2 R

p of �.�0/, i.e.,

˛.�0/ D j�.�0/j;

where �.�0/ D ��1.�0/; : : : ; �p.�0/

�T. The components of the vector �.�0/ are the

ratios of each standard error for a parameter to the corresponding nominal parametervalue. These ratios are dimensionless numbers warranting comparison even whenparameters have considerably different scales and units (e.g., in case of the HIV-model NT is on the order of 101, while k1 is on the order of 10�6, whereas incase of the CVS-model we have c` on the order 10�2 and Aexer

pesk on the order 102).A selection score ˛.�0/ near zero indicates lower uncertainty possibilities in theestimation, while large values of ˛.�0/ suggest that one could expect to findsubstantial uncertainty in at least some of the components of the estimates in anyparameter estimation attempt.

Let F0 be the Fisher information matrix corresponding to the parameter vectorsq 2 R

p0 and Fp the Fisher information matrix corresponding to the parametervectors in � 2 Sp . Then rank F0.q0/ D p0 implies that rank Fp.�0/ D p forany � 2 Sp , p < p0, i.e., if F0.q0/ is non-singular then also all Fp.�0/ are

3 Parameter Selection Methods in Inverse Problem Formulation 51

non-singular for all p < p0 and all �0 corresponding to a � 2 Sp . Moreover, ifrank F0.q0/ D p1 with p1 < p0, then rank F .�0/ < p for all p1 < p < p0 and all� 2 Sp .

On the basis of the considerations given above our algorithm proceeds as follows:

Selection Algorithm. Given p < p0 the algorithm considers all possible choicesof indices i1; : : : ; ip with 1 � i1 < � � � < ip � p0 in lexicographical ordering

starting with the first choice .i.1/1 ; : : : ; i

.1/p / D .1; : : : ; p/ and completes the

following steps:Initializing step: Set indsel D .1; : : : ; p/ and ˛sel D 1.Step k: For the choice .i

.k/1 ; : : : ; i

.k/p / compute r D rank F

�.q0/i

.k/1

; : : : ; .q0/i.k/p

/�.

If r < p, go to Step k C 1.If r D p, compute ˛k D ˛

�.q0/i

.k/1

; : : : ; .q0/i.k/p

/�.

If ˛k � ˛sel, go to Step k C 1.If ˛k < ˛sel, set indsel D .i

.k/1 ; : : : ; i

.k/p /, ˛sel D ˛k and

go to Step k C 1.Following the initializing step the algorithm performs

�p0

p

�steps and provides the

index vector indsel D �i�1 ; : : : ; i�

p

�which gives the sub-vector �� D �

qi�

1; : : : ; qi�

p

�T

such that the selection score ˛�.q0/i�

1; : : : ; .q0/i�

p

�is minimal among all possible

choices of sub-vectors in Sp . If rank Fp0 D p0 then the rank test in Step k can becancelled, of course.

3.4 Models

In the following, we shall illustrate the parameter selection ideas with resultsobtained by use of the subset selection algorithm described in the previous sectionfor two specific models. These models have a moderate number of parameters to beidentified yet are sufficiently complex to make a trial-error approach unfeasible.

3.4.1 A Mathematical Model for HIV Progression withTreatment Interruption

As our first illustrative example we use one of the many dynamic models for HIVprogression found in an extensive literature (e.g., see [1–4, 7, 20, 26, 49, 51, 58] andthe many references therein). For our example model, the dynamics of in-host HIVis described by the interactions between uninfected and infected type 1 target cells(T1 and T �

1 ) (CD4C T-cells), uninfected and infected type 2 target cells (T2 and T �2 )

52 H.T. Banks et al.

(such as macrophages or memory cells, etc.), infectious free virus VI , and immuneresponse E (cytotoxic T-lymphocytes CD8C) to the infection. This model, whichwas developed and studied in [1, 3] and later extended in subsequent efforts (e.g.,see [7]), is based on one suggested in [26], but includes an immune responsecompartment and dynamics as in [20]. The model equations are given by

PT1 D 1 � d1T1 � �1 � N�1.t/

�k1VI T1;

PT �1 D .1 � N�1.t/

�k1VI T1 � ıT �

1 � m1ET �1 ;

PT2 D 2 � d2T2 � .1 � f N�1.t//k2VI T2;

PT �2 D �

1 � f N�1.t/�k2VI T2 � ıT �

2 � m2ET �2 ;

PVI D �1 � N�2.t/

�103NT ı.T �

1 C T �2 / � cVI

� �1 � N�1.t/

�103k1T1VI � .1 � f N�1.t//103k2T2VI ;

PE D E C bE

�T �

1 C T �2

�

T �1 C T �

2 C Kb

E � dE.T �1 C T �

2 /

T �1 C T �

2 C Kd

E � ıEE;

(3.17)

together with an initial condition vector

�T1.0/; T �

1 .0/; T2.0/; T �2 .0/; VI .0/; E.0/

�T:

The differences in infection rates and treatment efficacy help create a low, butnon-zero, infected cell steady state for T �

2 , which is compatible with the ideathat macrophages or memory cells may be an important source of virus afterT-cell depletion. The populations of uninfected target cells T1 and T2 may havedifferent source rates i and natural death rates di . The time-dependent treatmentfactors N�1.t/ D �1u.t/ and N�2.t/ D �2u.t/ represent the effective treatment impactof a reverse transcriptase inhibitor (RTI) (that blocks new infections) and aprotease inhibitor (PI) (which causes infected cells to produce non-infectious virus),respectively. The RTI is potentially more effective in type 1 target cells (T1 and T �

1 )than in type 2 target cells (T2 and T �

2 ), where the efficacy is f N�1, with f 2 Œ0; 1.The relative effectiveness of RTIs is modeled by �1 and that of PIs by �2, whilethe time-dependent treatment function 0 � u.t/ � 1 represents therapy levels,with u.t/ D 0 for fully off and u.t/ D 1, for fully on. Although HIV treatment isnearly always administered as combination therapy, the model allows the possibilityof mono-therapy, even for a limited period of time, implemented by consideringseparate treatment functions u1.t/; u2.t/ in the treatment factors.

As in [1, 3], for our numerical investigations we consider a log-transformed andreduced version of the model. This transformation is frequently used in the HIVmodeling literature because of the large differences in orders of magnitude in statevalues in the model and the data and to guarantee non-negative state values as wellas because of certain probabilistic considerations (for further discussions see [3]).This results in the nonlinear system of differential equations

3 Parameter Selection Methods in Inverse Problem Formulation 53

Px1 D 10�x1

ln.10/

�1 � d110x1 � �

1 � N"1.t/�k110x510x1

�;

Px2 D 10�x2

ln.10/

��1 � N"1.t/

�k110x510x1 � ı10x2 � m110x610x2

�;

Px3 D 10�x3

ln.10/

�2 � d210x3 � �

1 � f N"1.t/�k210x510x3

�;

Px4 D 10�x4

ln.10/

��1 � f N"1.t/

�k210x510x3 � ı10x4 � m210x610x4

�;

Px5 D 10�x5

ln.10/

��1 � N"2.t/

�103NT ı

�10x2 C 10x4

� � c10x5

� �1 � N"1.t/

�103k110x110x5 � �

1 � f N"1.t/�103k210x310x5

�;

Px6 D 10�x6

ln.10/

�E C bE

�10x2 C 10x4

�

10x2 C 10x4 C Kb

10x6

� dE

�10x2 C 10x4

�

10x2 C 10x4 C Kd

10x6 � ıE10x6

�;

(3.18)

where the changes of variables are defined by

T1 D 10x1; T �1 D 10x2; T2 D 10x3; T �

2 D 10x4; VI D 10x5; E D 10x6:

The initial conditions for equations (3.18) are denoted by xi .t0/ D x0i , i D 1; : : : ; 6.

We note that this model has six state variables and the following 20 (in general,unknown) system parameters in the right-hand sides of equations (3.18)

1; d1; �1; k1; 2; d2; f; k2; ı; m1; m2; : : :

: : : �2; NT ; c; E; bE; Kb; dE; Kd ; ıE:

We may also consider the initial conditions as unknowns and thus we have 26unknown parameters which we collect in the parameter vector � ,

� D �x0

1 ; x02 ; x0

3 ; x04; x0

5 ; x06 ; 1; d1; �1; k1; 2; d2; f; k2; ı; m1; m2; : : :

: : : �2; NT ; c; E ; bE; Kb; dE; Kd ; ıE

�T:

A list of the parameters in the model equations along with their units is given belowin Table 3.1.

As reported in [1, 3], data to be used with this model in inverse or parameterestimation problems typically consisted of observations for T1 C T �

1 and V oversome extended time period. For the purpose of this paper we are only using thedata for T1 C T �

1 in case of patient # 4 which we depict in Fig. 3.1 together with

54 H.T. Banks et al.

Table 3.1 Parameters in the equations of the HIV model

Parameter Units Description

1 cells/(ml day) Production rate of type 1 target cellsd1 1/day Death rate of type 1 target cells�1 — Treatment efficacy of type 1 target cellsk1 ml/(virion day) Infection rate of type 1 target cells2 cells/(ml day) Production rate of type 2 target cellsd2 1/day Death rate of type 2 target cellsf — Reduction of treatment efficacy for type 2 target cellsk2 ml/(virion day) Infection rate of type 2 target cellsı 1/day Death rate of infected cellsm1 ml/(cell day) Immune-induced clearance rate for type 1 target cellsm2 ml/(cell day) Immune-induced clearance rate for type 2 target cells�2 — Treatment efficacy for type 2 target cellsNT virions/cell Virions produced per infected cellc 1/day Natural death rate of virusesE cells/(ml day) Production rate for immune effectorsbE 1/day Maximum birth rate for immune effectorsKb cells/ml Saturation constant for immune effector birthdE 1/day Maximum death rate for immune effectorsKd cells/ml Saturation constant for immune effector deathıE 1/day Natural death rate for immune effectors

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

Time (days)

Log(

CD

4+ T

−ce

lls/μ

l)

Patient #4

Fig. 3.1 Log-scaled data fyj g for CD4C T-cells of Patient #4 (crosses), and model output z.t /(solid curve) evaluated at the parameter estimates obtained in [1, 3]

the corresponding model output for the parameters identified in [1, 3]. Thus ourobservations are

f .ti I �0/ D log10

�10x1.ti I�0/ C 10x2.ti I�0/

�; (3.19)

where the nominal parameter vector �0 is given at the beginning of Sect. 3.5.1.

3 Parameter Selection Methods in Inverse Problem Formulation 55

While the inverse problem we are considering in this paper for the HIV-model is relatively “small” compared to many of those found in the literature,it still represents a nontrivial estimation challenge and is more than sufficientto illustrate the ideas and methodology we discuss in this presentation. Otherdifficult aspects (censored data requiring use of the Expectation Maximizationalgorithm as well as use of residual plots in attempts to validate the correctnessof choice of corresponding statistical models introduced and discussed in Sect. 3.2)of such inverse problems are discussed in the review chapter [8] and will not bepursued here.

3.4.2 A Model for the Reaction of the Cardiovascular Systemto an Ergometric Workload

As a second example to illustrate our methods we chose a model for cardiovascularfunction. The model was developed in order to describe the reaction of thecardiovascular system to a constant ergometric workload of moderate size. Thebuilding blocks of the model are the left ventricle, the arterial and venous systemiccompartments representing the systemic circuit as well as the right ventricle,arterial and venous pulmonary compartments representing the pulmonary circuit.The model is non-pulsatile and includes the baroreceptor loop as the essentialcontrol loop in the situation to be modeled. The feedback control which steers thesystem from the equilibrium corresponding to rest to the equilibrium correspondingto the imposed workload is obtained by solving a linear quadratic regulator problem.Furthermore the model includes a sub-model describing the so called autoregulationprocess, i.e., the control of local blood flow in response to local metabolic demands.The model equations are given as (for details see [44], [16, Chap. 1]):

PPas D 1

cas

�Q` � 1

Rs.Pas � Pvs/

�;

PPvs D 1

cvs

� 1

Rs.Pas � Pvs/ � Qr

�;

PPap D 1

cap

�Qr � 1

Rp.Pap � Pvp.Pas; Pvs; Pap//

�;

PS` D �`;

P�` D �˛`S` � �`�` C ˇ`H;

PSr D �r;

P�r D �˛rSr � �r�r C ˇrH;

PRs D 1

K

�Apesk

�Pas � Pvs

RsCa;O2 � M0 � �W

�� .Pas � Pvs/

�;

PH D u.t/

(3.20)

56 H.T. Banks et al.

with

Pvp D Pvp.Pas; Pvs; Pap/ WD 1

cvp

�Vtot � casPas � cvsPvs � capPap

�;

Q` D Hc`Pvp.Pas; Pvs; Pap/a`.H/S`

a`.H/Pas C k`.H/S`

;

Qr D HcrPvsar.H/Sr

ar.H/Pap C kr.H/Sr;

(3.21)

where

k`.H/ D e�.c`R`/�1td .H/ and a`.H/ D 1 � k`.H/;

kr.H/ D e�.crRr/�1td .H/ and ar.H/ D 1 � kr.H/:

(3.22)

For the duration td of the diastole we use Bazett’s formula (duration of the systole D =H 1=2) which implies

td D td .H/ D 1

H 1=2

� 1

H 1=2�

�: (3.23)

Introducing x D �Pas; Pvs; Pap; S`; �`; Sr; �r; Rs; H

�T 2 R9 system (3.20) can

be written as

Px.t/ D f .x.t/; W; �; u.t//;

where W D W rest D 0 (Watt) for t � 0 and W D W exer D 75 (Watt) for t > 0.Moreover, � is the parameter vector of the system. We distinguish two valuesfor each of the parameters Rp and Apesk, one for the resting situation and onefor the exercise situation. Consequently we have the parameters Rrest

p , Arestpesk and

Rexerp and Aexer

pesk instead of Rp and Apesk. The initial value for system (3.20) is theequilibrium xrest, which is computed from f .xrest; W rest; �; 0/ D 0. Analogouslyxexer is the equilibrium corresponding to the constant workload W exer (satisfyingf .xexer; W exer; �; 0/ D 0).

Let B D �0; : : : ; 0; 1

�T 2 R9, C D �

1; 0; : : : ; 0/ 2 R1�9 and denote by A.�/ D

@f .x;W exer;�;0/

@x

ˇˇxDxexer , the Jacobian of the right-hand side of system (3.20) at the

equilibrium xexer. The control u.t/ is obtained as the solution of the linear-quadraticregulator problem for the linear system

P�.t/ D A.�/�.t/ C Bu.t/; �.0/ D xrest � xexer; (3.24)

where we have set �.t/ D x.t/ � xexer, and the quadratic cost functional

J.u/ DZ 1

0

�q2

as.Pas.t/ � P exeras /2 C u.t/2

�dt: (3.25)

3 Parameter Selection Methods in Inverse Problem Formulation 57

Table 3.2 Parameters of the CVS-model

Parameter Units Description

cas l/mmHg Compliance of the arterial systemic compartmentcvs l/mmHg Compliance of the venous systemic compartmentcap l/mmHg Compliance of the arterial pulmonary compartmentcvp l/mmHg Compliance of the venous pulmonary compartmentc` l/mmHg Compliance of the relaxed left ventriclecr l/mmHg Compliance of the relaxed right ventricleVtot l Total blood volumeRp mmHg min/l Resistance in the peripheral region of the pulmonary circuitR` mmHg min/l Inflow resistance of the left ventricleRr mmHg min/l Inflow resistance of the right ventricle min1=2 Coefficient in Bazett’s formula (see (3.23))Ca;O2 1 O2-concentration in arterial systemic bloodK l Constant in the formula for the biochemical energy flow,

Mb D �K dCv;O2 =dt

Apesk mmHg min/l Constant in the formula relating peripheral systemicresistance and venous O2 concentration (Rs D ApeskCv;O2 /

M0 l/min Metabolic rate in the systemic tissue region correspondingto zero workload

� l/(min Watt) Coefficient of W in the differential equation for Rs

qas min�2(mmHg)�1 Weighting factor for Pas in the cost functional (3.25)˛` min�2 Coefficient of S` in the differential equation for S`

˛r min�2 Coefficient of Sr in the differential equation for Sr

ˇ` mmHg/min Coefficient of H in the differential equation for S`

ˇr mmHg/min Coefficient of H in the differential equation for Sr

�` min�1 Coefficient of PS` in the differential equation for S`

�r min�1 Coefficient of PSr in the differential equation for Sr

This functional reflects the fact that the baroreceptor loop, which is the basic controlloop for the situation we are considering here, generates the control using the arterialsystemic pressure Pas.t/.

According to the theory of the linear-quadratic control problem u.t/ is given by

u.t/ D �BTX�.t/ D �BTX.x.t/ � xexer/; t � 0; (3.26)

where X D X.�/ is the solution of the Riccati matrix equation XA.�/ CA.�/TX �XBBTX C C TC D 0. The feedback control (3.26) is also a stabilizing controlfor system (3.20), i.e., we have limt!1 x.t/ D xexer provided kx.0/ � xexerk2 issufficiently small.

The parameter vector of the system is

q D �c`; cr; cas; cvs; cap; cvp; R`; Rr; ˛`; ˛r; ˇ`; ˇr; �`; : : :

: : : �r; K; ; M0; �; Ca;O2 ; qas; Vtot; Rrestp ; Arest

pesk; Rexerp ; Aexer

pesk

�T 2 R25: (3.27)

Tables 3.2 and 3.3 contain the descriptions and units for the parameters q and thestate variables x, respectively, of the system.

58 H.T. Banks et al.

Table 3.3 The state variables and other variables of the CVS-model

Variable Unit Description

Pas mmHg Pressure in the arterial systemic compartmentPvs mmHg Pressure in the venous systemic compartmentPap mmHg Pressure in the arterial pulmonary compartmentPvp mmHg Pressure in the venous pulmonary compartmentS` mmHg Contractility of the left ventricle�` mmHg/min Time derivative of S`

Sr mmHg Contractility of the right ventricle�r mmHg/min Time derivative of Sr

Rs mmHg min/l Peripheral resistance in the systemic circuitH min�1 Heart rate

Q` l/min Cardiac output of the left ventricleQr l/min Cardiac output of the right ventricleW Watt Workload imposed on the test person

3.5 Results and Discussion

In this section we discuss some of our findings when we applied the SelectionAlgorithm presented in Sect. 3.3 to the HIV-model and the CVS-model.

3.5.1 The HIV-Model

As the nominal parameter vector we take the estimates obtained in [1, 3] forPatient # 4. More precisely, we assume that the error variance is �2

0 D 0:11, and thatthe nominal parameter values (for description and units see Table 3.1) are given as:

x01 D log10.1:202eC3/; x0

2 D log10.6:165eC1/; x03 D log10.1:755eC1/;

x04 D log10.6:096e�1/; x0

5 D log10.9:964eC5/; x06 D log10.1:883e�1/;

1 D 4:633; d1 D 4:533 � 10�3; �1 D 6:017 � 10�1;

k1 D 1:976 � 10�6; 2 D 1:001 � 10�1; d2 D 2:211 � 10�2;

f D 5:3915 � 10�1; k2 D 5:529 � 10�4; ı D 1:865 � 10�1;

m1 D 2:439 � 10�2; m2 D 1:3099 � 10�2; �2 D 5:043 � 10�1;

NT D 1:904 � 101; c D 1:936 � 101; E D 9:909 � 10�3;

bE D 9:785 � 10�2; Kb D 3:909 � 10�1; dE D 1:021 � 10�1;

Kd D 8:379 � 10�1; ıE D 7:030 � 10�2:

3 Parameter Selection Methods in Inverse Problem Formulation 59

In Fig. 3.1 above we depicted the log-scaled longitudinal observations fyig on thenumber of CD4C T-cells and the model output z.tj / D f .tj ; �0/, j D 1; : : : ; N ,evaluated at the nominal parameter vector and given in (3.19).

It is assumed that the following parameters are always fixed at the values givenabove:

x03 ; x0

4 ; and x06 :

As reported in [1, 3], simulation and relative sensitivity studies revealed that thevalidated models are not sensitive to these initial conditions and following [3] wetherefore fix them. In other words, the parameters to be selected for estimation willalways constitute a sub-vector of

q D �x0

1 ; x02; x0

5 ; 1; d1; �1; k1; 2; d2; f; k2; ı; : : :

: : : m1; m2; �2; NT ; c; E; bE; Kb; dE; Kd ; ıE

� 2 R23: (3.28)

Let F23 denote the Fisher information matrix of system (3.18) for the 23parameters of q as given in (3.28) at their nominal values q0 as given above. Thenwe have

cond F23.q0/ D 1:712 � 1024;

i.e., F23.q0/ is non-singular, but ill-conditioned. Since F23.q0/ is non-singular, theFisher information matrix for any sub-vector � of q at the corresponding nominalparameter values is also non-singular. Consequently the regularity check for theFisher information matrix in the subset selection algorithm can be deleted in case ofthe HIV-model.

In [1, 3], the authors estimate the parameter vector

� D �x0

1; x02 ; x0

5 ; 1; d1; �1; k1; �2; NT ; c; bE

�T 2 R11: (3.29)

The selection score for this parameter vector is ˛.�0/ D 4:611 � 103. In Table 3.4we display, for the five selections of sub-vector � 2 R

11 of q with the minimalselection scores, the condition numbers of the corresponding Fisher informationmatrices and the selection scores. In addition we also show the selection � 2 R

11

with the maximal selection score. As we can see, the selection score values rangefrom 2:881�101 to 2:340�105 for the

�2311

� D 1;352;078 different parameter vectorsin R

11 which can be selected from the 23 parameters in (3.28).As we also can see from Table 3.4 that the selection algorithm chooses

most of the parameters in the vector (3.29). For instance, the sub-vector .x05 ; 1;

d1; �1; �2/ of (3.29) appears in every one of the top five parameter vectors dis-played in Table 3.4. In fact the top five parameter vectors have the sub-vector.x0

5 ; 1; d1; �1; 2; d2; k2; ı; �2/ 2 R9 in common and differ only by one or two

of the parameters x01 , x0

2 , k1, c, and NT . Use of the subset selection algorithmdiscussed here (had it been available) might have proved valuable in the effortsreported in [1, 3].

60 H.T. Banks et al.

Table 3.4 The top five parameter vectors obtained with subset selection algorithm for p D 11.For each parameter vector � the condition number .F .�0// of the Fisher information matrix andthe selection score ˛.�0/ are displayed. The next two lines show .F .�0// and ˛.�0/ for the sub-vector � 2 R

9 which is common to the top five parameter vectors and for the optimal parametervector in R

9. The last line presents the sub-vector in R11 with the largest selection score

Parameter vector � .F .�0// ˛.�0/

.x01 ; x0

5 ; 1; d1; �1; 2; d2; k2; ı; �2; NT / 9.841 � 1010 2.881 � 101

.x01 ; x0

5 ; 1; d1; �1; 2; d2; k2; ı; �2; c/ 9.845 � 1010 2.883 � 101

.x01 ; x0

5 ; 1; d1; �1; k1; 2; d2; k2; ı; �2/ 4.388 � 1016 2.896 � 101

.x02 ; x0

5 ; 1; d1; �1; 2; d2; k2; ı; �2; NT / 9.235 � 1010 2.904 � 101

.x02 ; x0

5 ; 1; d1; �1; 2; d2; k2; ı; �2; c/ 9.241 � 1010 2.906 � 101

.x05 ; 1; d1; �1; 2; d2; k2; ı; �2/ 9.083 � 1010 2.193 � 101

.x01 ; x0

5 ; 1; d1; k1; d2; k2; ı; �2/ 4.335 � 1015 1.050 � 101

.d2; k2; ı; m2; NT ; E; bE ; Kb; dE ; Kd ; ıE/ 7.247 � 1017 2.340 � 105

In Fig. 3.2a, we depict the minimal selection score as a function of the numberof parameters. Table 3.5 contains the values of the corresponding selection scores.Figure 3.2b is a semilog plot of Fig. 3.2a, i.e., it displays the logarithm of theselection score as a function of the number of parameters. Figure 3.2b, suggeststhat parameter vectors with more than 13 parameters might be expected to havelarge uncertainty when estimated from observations, because the minimal selectionscore is already larger than 100. Figure 3.2b also depicts the regression line, whichfits the logarithm of the selection score. From this linear regression we conclude theselection score ˛min.p/ grows exponentially with the number p of parameters to beestimated. More precisely, we find

˛min.p/ 0:01133e0:728p; p D 1; : : : ; 22: (3.30)

Computing the minimal selection scores for p D 1; : : : ; 22 requires to consider allpossible choices of sub-vectors of (3.28) in R

p , p D 1; : : : ; 22, i.e., to considerP22iD1

�23

i

� D 8;388;605 cases. If we determine the regression line only using theminimal selection scores for p D 1; 2; 3; 20; 21; 22 we obtain

˛min.p/ 0:01441e0:710p; p D 1; : : : ; 22: (3.31)

Computing the minimal selection scores needed for (3.31) requires to consideronly

�231

� C �232

� C �233

� C �2320

� C �2321

� C �2322

� D 2��

231

� C �232

� C �233

�� D 4;094 cases.In Fig. 3.2a, we show the curves given by (3.30) and (3.31), whereas in Fig. 3.2bthe corresponding regression lines are depicted. Table 3.6 shows the time it takeson a laptop computer with an Intel c� CoreTM2 Duo processor using a MATLABprogramm to compute ˛min.p/, p D 1; : : : ; 22, once the Fisher information matrixF23.q0/ for the nominal parameter vector q0 has been computed. In order toreduce computational efforts it is important to observe that for any selection � of

3 Parameter Selection Methods in Inverse Problem Formulation 61

0 5 10 15 20 250

2

4

6

8

10

12x 104

p

αm

in(p

)

0 5 10 15 20 25−2

−1

0

1

2

3

4

5

6

p

log

10( α

min

(p))

a

b

Fig. 3.2 (a) Minimal selection scores (crosses) and exponential approximations (3.30) (grey solidline) respectively (3.31) (grey dashed line) versus the number of parameters p. (b) Logarithm ofminimal selection scores (crosses) and regression lines corresponding to (3.30) (gray solid line)respectively to (3.31) (grey dashed line) versus number of parameters p

Table 3.5 Minimal selection scores ˛min.p/ for sub-vectors in Rp of (3.28), p D 1; 2; : : : ; 22

p 1 2 3 4 5 6 7 8

˛min.p/ 0.0340 0.0523 0.1203 0.1679 9.3796 0.6511 1.1375 2.4166

p 9 10 11 12 13 14 15 16

˛min.p/ 10.503 17.482 28.81 92.91 243.34 336.77 566.86 2,274.3

p 17 18 19 20 21 22

˛min.p/ 3,372.4 5,047.9 9,664.4 16,585 51,631 95,128

62 H.T. Banks et al.

Table 3.6 Time for computing � 2 Rp with the minimal selection score on a laptop

computer, once the Fisher information matrix F23.q0/ for the HIV-model has beencomputed

p 1 2 3 4 5 6 7 8

time (s) 0.054 0.015 0.075 0.472 1.66 5.52 14.05 29.81

p 9 10 11 12 13 14 15 16

time (s) 53.49 80.02 100.4 105.8 96.74 74.01 47.15 24.09

p 17 18 19 20 21 22

time (s) 10.69 3.64 1.02 0.211 0.034 0.0041

0 1 2 3 4 5 6 7 8 9 10 110

5

10

15

20

25

30

35

40

p

αm

in(p

)

Fig. 3.3 Minimal selection scores (crosses) and exponential approximations (3.30) (grey solidline) respectively (3.31) (grey dashed line) for p D 1; : : : ; 11

p parameters from the set of p0 parameters the corresponding Fisher informationmatrix F .�0/ (�0 being the vector of nominal values of the components of �) isobtained from F23.q0/ simply by selecting the rows and columns corresponding tothe components of � .

If the condition number of the Fisher information matrix corresponding to theoriginal parameter set is becoming very large, then the approximation of ˛min.p/ byan exponential function might be possible only for p D 1; : : : ; p� with p� < p0.See the results for the CVS-model in Sect. 3.5.2.

Figure 3.3 is the same as Fig. 3.2a, but for p D 1; : : : ; 11. The curves (3.30)respectively (3.31) can be used to determine p such that the selection score ˛min.p/

is smaller than a given upper bound. From Fig. 3.3 we can see that in order to have

3 Parameter Selection Methods in Inverse Problem Formulation 63

10−2 10−1 100 101 102 103 104 105100

105

1010

1015

1020

1025

12 3

4 56

7 8 9

1011

12

13

1415

1617

1819 20 21 22

Fig. 3.4 Condition number .Fp.�0// versus minimal selection score ˛min.p/ for the HIV-model,where � 2 R

p is the sub-vector of (3.28) with the minimal selection score, p D 1; : : : ; 22. Bothaxes are in logarithmic scale

Table 3.7 The top five parameter vectors � 2 R11 selected according the criterion of minimal

condition number for the Fisher information matrix

Parameter vector � .F .�0// ˛.�0/

.x01 ; x0

2 ; 1; �1; f; m1; m2; NT ; E; dE; ıE/ 1:735 � 106 1:908 � 103

.x01 ; x0

2 ; 1; �1; f; m1; m2; NT ; E; bE; ıE/ 1:738 � 106 1:897 � 103

.x01 ; x0

2 ; 1; �1; f; m1; m2; c; E; dE; ıE/ 1:744 � 106 1:908 � 103

.x01 ; x0

2 ; 1; �1; f; m1; m2; c; E; bE ; ıE/ 1:747 � 106 1:896 � 103

.x02 ; x0

5 ; 1; �1; f; m1; m2; c; E; dE; ıE/ 1:788 � 106 1:910 � 103

˛min.p/ < 5 we should choose p � 8, which is correct according to each of thetwo curves (see Table 3.5). In order to have ˛min < 10 the curves suggest p � 9,which is not quite correct, because according to Table 3.5 we have ˛min.9/ D 10:5,so that we should choose p � 8. In Fig. 3.4, we graph (in logarithmic scales)the condition number .Fp.�0// of the corresponding Fisher information matrixversus the smallest selection score ˛min.p/ for p D 1; : : : ; 22. It is clear fromFig. 3.4 that the condition numbers for the Fisher information matrix correspondingto the selected parameter vector � 2 R

p does not show a monotone behavior withrespect to p. We could also determine the selection of parameters according to thecriterion of minimal condition number for the corresponding Fisher informationmatrix. In Table 3.7, we present the best five selections � 2 R

11 according to thiscriterion together with the condition numbers of the Fisher information matrix andthe corresponding selection scores.

In Table 3.8 we examine the effect that removing parameters from an estimationhas in uncertainty quantification. The coefficient of variation (CV) is shown for eachparameter (see (3.16)). Five cases are considered:

64 H.T. Banks et al.

Table 3.8 Coefficient of variation (CV) of the parameter vectors for the HIV-model as listed above

CV for

Parameter �.18/ � .5;1/ � .5;2/ � .5;3/ � .5;4/

x01 — — 4.09 � 10�2 — —

x02 3.80 — — — —

x05 1.58 � 101 3.43 � 10�1 — — —

1 8.19 � 10�1 3.56 � 10�1 1.13 � 10�1 — —

d1 9.39 � 10�1 3.94 � 10�1 — — —

�1 1.26 � 102 — — 8.49 —

k1 7.67 � 102 8.17 � 10�2 9.57 � 10�2 — —

2 4.74 � 101 — — 9.99 —

d2 4.62 � 101 — — — —

f 2.51 � 102 — — — —

k2 7.53 � 102 — — — —

ı — — 3.29 � 10�1 — —

m1 2.29 � 103 — — 9.85 � 102 2.10 � 101

�2 1.63 � 102 1.11 � 10�1 9.33 � 10�2 6.34 —

c 7.74 � 102 — — — —

E 2.18 � 103 — — 9.83 � 102 —

bE — — — — 1.22 � 104

Kb 2.55 � 103 — — — 4.62 � 103

dE 4.56 � 102 — — — 1.16 � 104

Kd 1.98 � 103 — — — 4.40 � 103

ıE 1.72 � 103 — — — —

˛.�/ 5.05 � 103 6.47 � 10�1 3.75 � 10�1 1.39 � 103 1.80 � 104

(i) The parameter vector

�.18/ D �x0

2 ; x05 ; 1; d1; �1; k1; 2; d2; f; k2; m1; �2; c; E ; dE; Kd ; ıE

�T;

which is the sub-vector in R18 with the minimal selection score.

(ii) The parameter vector

�.5;1/ D �x0

5; 1; d1; k1; �2

�T;

which is the sub-vector of �.18/ in R5 with the minimal selection score.

(iii) The parameter vector

�.5;2/ D �x0

1; 1; k1; ı; �2

�T;

which is the sub-vector in R5 of q as given by (3.28) with the minimal selection

score.

3 Parameter Selection Methods in Inverse Problem Formulation 65

(iv) The parameter vector

�.5;3/ D ��1; 2; m1; �2; E

�T;

which is the sub-vector of �.18/ in R5 with the maximal selection score.

(v) The parameter vector

�.5;4/ D �m1; bE; Kb; dE; Kd

�T;

which is the sub-vector in R5 of q as given by (3.28) with the maximal selection

score.

We see that here are substantial improvements in uncertainty quantificationwhen considering �.5;1/ instead of �.18/. However, just taking a considerablylower dimensional sub-vector of �.18/ in R

5 does not lead necessarily to a drasticimprovement of the estimate.

3.5.2 The CVS-Model

For this model we take as the nominal parameters the following estimates obtainedin [44] using data obtained at bicycle ergometer tests (for a description and units seeTable 3.2):

c` D 0:02305; cr D 0:04413; cas D 0:01016;

cvs D 0:6500; cap D 0:03608; cvp D 0:1408;

R` D 0:2671; Rr D 0:04150; ˛` D 30:5587;

˛r D 28:6785; ˇ` D 25:0652; ˇr D 1:4132;

�` D �1:6744; �r D �1:8607; K D 16:0376;

D 0:05164; M0 D 0:35; � D 0:011;

Ca;O2 D 0:2; qas D 163:047; Vtot D 5:0582;

Rrestp D 1:5446; Arest

pesk D 177:682; Rexerp D 0:3165;

Aexerpesk D 254:325:

For the estimates given above measurements for Pas, H and Q` were available. Thesampling times tj for the measurements of Pas and H were uniformly distributedon the time interval from 0 to 10 min with tj C1 � tj D 2 s, i.e., 601 measurementsfor Pas and H . The measurements for Q` were obtained by Doppler echo-cardiography and consequently were much less frequent (only 20 measurements)and also irregularly distributed. In the following we shall consider Pas as the only

66 H.T. Banks et al.

0 1 2 3 4 5 6 7 8 9 1090

95

100

105

110

115

120

125

130

135

140

time (min)

Pas

(m

mH

g)

Fig. 3.5 The Pas-component of the solution of system (3.20) with the nominal parameter valuesand the Pas-measurements

Table 3.9 The equilibria xrest and xexer for the CVS-model

Variable Pas Pvs Pap Pvp S` �` Sr �r Rs H

xrest 105.595 4.277 12.474 5.367 64.675 0 3.886 0 22.020 78.85

xexer 122.115 3.595 10.441 7.844 88.092 0 5.293 0 14.445 107.4

measured output of the system, i.e., we have f .t I �/ D Pas.t I �/. The variance ofthe measurement errors was roughly estimated to be �2

0 D 10 (Fig. 3.5).The equilibria xrest and xexer are given in Table 3.9 (for units see Table 3.2).As for the HIV-model also in case of the CVS-model the Fisher information

matrix F25.q0/ at the nominal values of the parameters (3.27) is non-singular, buthighly ill-conditioned:

cond F25.q0/ D 2:5755 � 1031:

Therefore we can also delete the regularity test for the Fisher information matrixin the selection algorithm in case of the CVS-model. In Fig. 3.7, we show theminimal selection scores for p D 3; : : : ; 20, whereas in Table 3.10 we list theminimal selection scores ˛min.p/, p D 1; : : : ; 24. Table 3.10 clearly shows the effectof the extreme ill-conditioning of the Fisher information matrix F25. We see anextreme increase of the selection score from 3 �105 at p D 21 to 2:4 �1031 at p D 22.This is also reflected in Fig. 3.6 which clearly shows that there is no reasonableapproximation of log

�˛min.p/

�, p D 1; : : : ; 24, by a regression line. However,

a regression line makes sense for p D 1; : : : ; 21 as can be see from Fig. 3.6. In

3 Parameter Selection Methods in Inverse Problem Formulation 67

Table 3.10 Minimal selection scores ˛min.p/, p D 1; : : : ; 24, for the CVS-model

p 1 2 3 4 5 6

˛min.p/ 9.397 � 10�4 2.042 � 10�2 5.796 � 10�2 1.096 � 10�1 2.315 � 10�1 3.299 � 10�1

p 7 8 9 10 11 12

˛min.p/ 4.340 � 10�1 6.251 � 10�1 8.018 � 10�1 1.084 1.723 3.238

p 13 14 15 16 17 18

˛min.p/ 6.336 17.13 37.31 59.37 2.224 � 102 4.301 � 102

p 19 20 21 22 23 24

˛min.p/ 1.015 � 103 1.265 � 105 3.038 � 105 2.396 � 1031 1.834 � 1034 2.459 � 10108

0 5 10 15 20 25−20

0

20

40

60

80

100

120

p

log

10(α

min

(p))

Fig. 3.6 Logarithm of the minimal selection scores, p D 1; : : : ; 24, for the CVS-model

Fig. 3.7, we depict the minimal selection scores ˛min.p/, p D 2; : : : ; 19, togetherwith the approximating exponential functions

˛min.p/ 0:00670e0:580p; p D 1; : : : ; 19; (3.32)

obtained from the regression line for p D 1; : : : ; 19 (solid grey line) and

˛min.p/ 0:00799e0:610p; p D 1; : : : ; 19; (3.33)

obtained for p D 2; 3; 4; 17; 18; 19 (dashed grey line). Table 3.11 for the CVS-model is the analogue to Table 3.6 (for the HIV-model) concerning computing times.

In Table 3.12, we present the sub-vector � 2 R10 of q given by (3.27) with

the minimal selection score together with the coefficients of variation. In Fig. 3.8,we depict the classical sensitivities for the chosen sub-vector � 2 R

10 (black solidlines) and the sensitivities for the other 15 parameters (grey dashed lines). We seethat the algorithm chooses not only the parameters with large sensitivities, but also

68 H.T. Banks et al.

2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

p

αm

in(p

)

2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

3

4

p

log

10(α

min

(p))

Fig. 3.7 Minimal selection scores ˛min.p/ (upper panel) and log10

�˛min.p/

�(lower panel), p D

2; : : : ; 19, for the CVS-model

Table 3.11 Time for computing � 2 Rp with the minimal selection score on a laptop computer,

once the Fisher information matrix F25.q0/ for the CVS-model has been computed

p 1 2 3 4 5 6 7 8 9

time (s) 0.0038 0.018 0.104 0.564 2.364 7.848 21.24 46.60 88.31

p 10 11 12 13 14 15 16 17 18time (s) 139.7 188.3 211.1 208.9 172.9 123.1 71.41 36.10 14.92

p 19 20 21 22 23 24

time (s) 5.137 1.382 0.329 0.093 0.0645 0.0005

3 Parameter Selection Methods in Inverse Problem Formulation 69

Table 3.12 Coefficients of variance for the sub-vector � 2 R10 with the minimal selection score

� 2 R10 ˛` ˛r ˇ` �` �r � Ca;O2 qas Vtot Rrest

p

CV 0.086 0.183 0.435 0.353 0.559 0.166 0.235 0.485 0.368 0.240

˛min.10/ 1.086

0 1 2 3 4 5 6 7 8 9 10−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (min)

sens

itivi

ties

0 1 2 3 4 5 6 7 8 9 10−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

time (min)

sens

itivi

ties

Fig. 3.8 Classical sensitivities of Pas with respect to the parameters of the selection � D�˛`; ˛r; ˇ`; �`; �r; �; Ca;O2 ; qas; Vtot;R

restp

�T 2 R10 which provides the minimal selection score

(black solid lines) and classical sensitivities of P as with respect to the remaining 15 parameters(dashed grey lines). The lower panel is a blow-up of the part of the upper panel with sensitivitiesbetween �0:1 and 0:1

parameters with rather small sensitivities. In this context one has to observe thatthe subset selection algorithm chooses parameters which can be jointly identifiedwith a given accuracy (in terms of asymptotic standard errors), whereas classicalsensitivities characterize the sensitivity of the measurable output with respect to asingle parameter.

70 H.T. Banks et al.

3.6 Concluding Remarks

As we have noted, inverse problems for complex system models containing a largenumber of parameters are difficult. There is great need for quantitative methodsto assist in posing inverse problems that will be well formulated in the senseof the ability to provide parameter estimates with quantifiable small uncertaintyestimates. We have introduced and illustrated use of such an algorithm that requiresprior local information about ranges of admissible parameter values and initialvalues of interest along with information on the error in the observation processto be used with the inverse problem. These are needed in order to implement thesensitivity/Fisher matrix based algorithm.

Because sensitivity of a model with respect to a parameter is fundamentallyrelated to the ability to estimate the parameter, and because sensitivity is a localconcept, we observe that the pursuit of a global algorithm to use in formulatingparameter estimation or inverse problems is most likely a quest that will gounfulfilled.

Acknowledgements This research was supported in part by Grant Number R01AI071915-07from the National Institute of Allergy and Infectious Diseases and in part by the Air Force Office ofScientific Research under grant number FA9550-09-1-0226. The content is solely the responsibilityof the authors and does not necessarily represent the official views of the NIAID, the NIH or theAFOSR.

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